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Ma 322: Biostatistics Homework Assignment 9 Prof. Wickerhauser Due Friday, April 7th, 2017 Read Chapter 15, “ANOVA and Regression,” pages 263–287 of our text. NOTE: It should be possible to cut and paste the data from this document into a text file, or into an R variable by use of the scan() function. 1. (a) How many 2 × 2 contingency tables are there with row sums (2, 5) and column sums (3, 4)? (Hint: Write down all the solutions.) (b) Assuming that the rows and columns are independent, compute the exact hypergeometric probability of each 2 × 2 contingency table in part a. 2. The following data are frequencies of bats found with and without rabies in two different geographic areas: Area E W With rabies 18 11 Without rabies 112 139 (a) Using the Yates-corrected χ2 test at the α = 0.05 significance level, test H0 : the incidence of rabies is the same in both areas. (b) Use the Fisher exact test at the 0.05 level to test if the E population bats are more likely to have rabies than those in the W population. 3. A follow-on study was performed on the same bats data, similar to that of Problem 2 but with the additional tabulation of gender: Area E W With Male 6 9 rabies Female 12 2 Without rabies Male Female 49 63 84 55 (a) Test for mutual independence at the α = 0.1 significance level. (b) Test for partial independence at the α = 0.05 significance level. 1 4. The following fake data mimics a study of amino acids in six imaginary species of millipedes: Alanine concentration in millipede hæmolymph (mg/100 ml) Species 1 21.5 19.6 20.9 22.8 Species 2 14.5 17.4 15.0 17.8 Species 3 16.0 20.3 18.5 19.3 Species 4 14.8 15.6 13.5 16.4 Species 5 12.1 11.4 12.7 14.5 Species 6 14.4 14.7 13.8 12.0 (a) Test, at the α = 0.05 significance level, the hypothesis H0 : There is no difference in mean alanine concentration among the species. Use one-factor ANOVA. (b) Test, at the α = 0.05 significance level, the hypothesis H0 : There is no difference in mean alanine concentration between species A and B. Use pairwise t-tests for every pair A, B. (c) Test, at the α = 0.05 significance level, the hypothesis H0 : There is no difference in mean alanine concentration between species A and B. Use Tukey’s HSD test for every pair A, B. 5. Test for all factor and interaction effects in the following 3 × 2 fixed-effects analysis of variance with equal replication: Response a1 ---------b1 b2 ---- ---34.1 35.6 36.9 36.3 33.2 34.7 35.1 35.8 34.8 36.0 to Factors A and B a2 a3 ------------------b1 b2 b1 b2 ---- ------- ---38.6 40.3 41.0 42.1 39.1 41.3 41.4 42.7 41.3 42.7 43.0 43.1 41.4 41.9 43.4 44.8 40.7 40.8 42.2 44.5 6. Test for all factor and interaction effects in the following 4 × 3 × 2 fixed-effects analysis of variance, where ai is the level of factor A, bi is the level of factor B, and ci is the level of factor C. a1 ------------b1 b2 b3 --- --- --- Response to Factors A, B and C a2 a3 ------------------------b1 b2 b3 b1 b2 b3 --- --- ----- --- --- a4 ------------b1 b2 b3 --- --- --- 4.1 4.3 4.5 3.8 4.6 4.9 4.2 4.5 3.7 3.9 4.1 4.5 4.9 4.6 5.3 5.0 5.2 5.6 5.8 5.4 4.7 4.7 5.0 4.5 5.0 5.4 5.7 5.3 6.1 6.2 6.5 5.7 5.5 5.9 5.6 5.0 3.9 3.3 3.4 3.7 4.4 4.3 4.7 4.1 3.7 3.9 4.0 4.4 4.8 4.5 5.0 4.6 5.6 5.8 5.4 6.1 5.0 5.2 4.6 4.9 4.9 5.5 5.5 5.3 5.9 5.3 5.5 5.7 5.0 5.4 4.7 5.1 6.0 5.7 5.5 5.7 6.0 6.3 5.7 5.9 6.1 5.3 5.5 5.8 4.1 3.9 4.3 4.0 4.9 4.7 4.9 5.3 4.3 4.1 3.8 4.7 c1: c2: 2